Drillers: Once you unlock the Experimental Plasma Charger (EPC), this can be used to mine resources. You charge a full shot, fire, and fire a small shot to burst it. Getting the timing takes a bit, but it's very useful. Especially for ceiling mining.
Scouts: Light. The. Rooms. That's your main function, and it's so helpful. Also, fetch minerals. You can do a grappling hook (letting go before you hit the destination) to momentum fling yourself and then finish with a power attack to make a landing area in a wall to mine from. This is a big time saver. Opinion - scouts should call the MULE sparingly. They can get to the MULE way faster than it can get to the, and you're also delaying all the other, slower moving players from using it.
Gunners: Barrier is so good. Save it for when you really need it (especially for picking up others. Ziplines are like crappy grappling hooks that everyone can use. Making them cross over each other in x patterns can be helpful so you're able to go faster (on a downward slope) both directions.
Engineers: I may not be the best person to ask. I find Engineers very tedious and boring. Too much gun hammering and refilling of gun ammo, and no particularly useful mobility tools (the platform gun is the meh-est of the mobility tools, in my opinion). Other may have differing opinions on the matter though.
Welcome to DRG! Rock and Stone, Brother (or sister or sibling)!